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SPRINGFIELD PUBLIC SCHOOLS
GEOMETRY Geometry will require students to explore complex geometric situations and deepen
their explanations of geometric relationships, moving towards formal mathematical
arguments. Emphasis is placed on using deductive reasoning in the analysis of
topics such as parallel lines, circles, polygon congruence, similarity, area, volume,
and probability. Content will include both coordinate and transformational
geometry. Prerequisite: C or better in Algebra I or Algebra II.
1. Congruence, Proof, and Constructions - In previous grades, students were
asked to draw triangles based on given measurements. They also have prior
experience with rigid motions: translations, reflections, and rotations and have used
these to develop notions about what it means for two objects to be congruent. In
this unit, students establish triangle congruence criteria, based on analyses of rigid
motions and formal constructions. They use triangle congruence as a familiar
foundation for the development of formal proof. Students prove theorems—using a
variety of formats—and solve problems about triangles, quadrilaterals, and other
polygons. They apply reasoning to complete geometric constructions and explain
why they work.
2. Similarity, Proof, and Trigonometry - Students apply their earlier experience
with dilations and proportional reasoning to build a formal understanding of
similarity. They identify criteria for similarity of triangles, use similarity to solve
problems, and apply similarity in right triangles to understand right triangle
trigonometry, with particular attention to special right triangles and the Pythagorean
theorem. Students develop the Laws of Sines and Cosines in order to find missing
measures of general (not necessarily right) triangles, building on students’ work
with quadratic equations done in the first course. They are able to distinguish
whether three given measures (angles or sides) define 0, 1, 2, or infinitely many
triangles.
3. Extending to Three Dimensions - Students’ experience with two-dimensional
and three-dimensional objects is extended to include informal explanations of
circumference, area and volume formulas. Additionally, students apply their
knowledge of two-dimensional shapes to consider the shapes of cross-sections
and the result of rotating a two-dimensional object about a line.
4. Connecting Algebra and Geometry through Coordinates - Building on their
work with the Pythagorean theorem in 8th grade to find distances, students use a
rectangular coordinate system to verify geometric relationships, including
properties of special triangles and quadrilaterals and slopes of parallel and
perpendicular lines, which relates back to work done in the first course. Students
continue their study of quadratics by connecting the geometric and algebraic
definitions of the parabola.
5. Circles With and Without Coordinates - In this unit students prove basic
theorems about circles, such as a tangent line is perpendicular to a radius,
inscribed angle theorem, and theorems about chords, secants, and tangents
dealing with segment lengths and angle measures. They study relationships among
segments on chords, secants, and tangents as an application of similarity. In the
Cartesian coordinate system, students use the distance formula to write the
equation of a circle when given the radius and the coordinates of its center. Given
an equation of a circle, they draw the graph in the coordinate plane, and apply
techniques for solving quadratic equations, which relates back to work done in the
first course, to determine intersections between lines and circles or parabolas and
between two circles.
6. Applications of Probability - Building on probability concepts that began in the
middle grades, students use the languages of set theory to expand their ability to
compute and interpret theoretical and experimental probabilities for compound
events, attending to mutually exclusive events, independent events, and conditional
probability. Students should make use of geometric probability models wherever
possible. They use probability to make informed decisions.
MATHEMATICAL PRACTICES
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Congruence
Experiment with transformations in the plane.
G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel
line, and line segment, based on the undefined notions of point,
line, distance along a line, and distance around a circular arc.
G.CO.2 Represent transformations in the plane using, e.g., transparencies
and geometry software; describe transformations as functions that
take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to
those that do not (e.g., translation versus horizontal stretch).
G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon,
describe the rotations and reflections that carry it onto itself.
G.CO.4 Develop definitions of rotations, reflections, and translations in
terms of angles, circles, perpendicular lines, parallel lines, and line
segments.
G.CO.5 Given a geometric figure and a rotation, reflection, or translation,
draw the transformed figure using, e.g., graph paper, tracing paper,
or geometry software. Specify a sequence of transformations that
will carry a given figure onto another.
Understand congruence in terms of rigid motions.
G.CO.6 Use geometric descriptions of rigid motions to transform figures
and to predict the effect of a given rigid motion on a given figure;
given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
G.CO.7 Use the definition of congruence in terms of rigid motions to show
that two triangles are congruent if and only if corresponding pairs
of sides and corresponding pairs of angles are congruent.
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and
SSS) follow from the definition of congruence in terms of rigid
motions.
Prove geometric theorems.
G.CO.9 Prove theorems about lines and angles. Theorems include: vertical
angles are congruent; when a transversal crosses parallel lines,
alternate interior angles are congruent and corresponding angles
are congruent; points on a perpendicular bisector of a line segment
are exactly those equidistant from the segment’s endpoints.
G.CO.10 Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180°; base angles of isosceles
triangles are congruent; the segment joining midpoints of two sides
of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.
G.CO.11 Prove theorems about parallelograms. Theorems include: opposite
sides are congruent, opposite angles are congruent, the diagonals
of a parallelogram bisect each other, and conversely, rectangles
are parallelograms with congruent diagonals.
Make geometric constructions.
G.CO.12 Make formal geometric constructions with a variety of tools and
methods (compass and straightedge, string, reflective devices,
paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an
angle; constructing perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon
inscribed in a circle.
Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
G.SRT.1 Verify experimentally the properties of dilations given by a center
and a scale factor:
a.
A dilation takes a line not passing through the center of the
dilation to a parallel line, and leaves a line passing through
the center unchanged.
b.
The dilation of a line segment is longer or shorter in the ratio
given by the scale factor.
G.SRT.2 Given two figures, use the definition of similarity in terms of
similarity transformations to decide if they are similar; explain using
similarity transformations the meaning of similarity for triangles as
the equality of all corresponding pairs of angles and the
proportionality of all corresponding pairs of sides.
G.SRT.3 Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar.
SPRINGFIELD PUBLIC SCHOOLS
GEOMETRY Prove theorems involving similarity.
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel
to one side of a triangle divides the other two proportionally, and
conversely; the Pythagorean Theorem proved using triangle
similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve
problems and to prove relationships in geometric figures.
G.MG.2
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.6 Understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
G.SRT.7 Explain and use the relationship between the sine and cosine of
complementary angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve
right triangles in applied problems.
Understand independence and conditional probability.
S.CP.1
Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes, or
as unions, intersections, or complements of other events (“or,”
“and,” “not”).
S.CP.2
Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are
independent.
S.CP.3
Understand the conditional probability of A given B as P(A and
B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of
A, and the conditional probability of B given A is the same as the
probability of B.
S.CP.4
Construct and interpret two-way frequency tables of data when two
categories are associated with each object being classified. Use
the two-way table as a sample space to decide if events are
independent and to approximate conditional probabilities. For
example, collect data from a random sample of students in your
school on their favorite subject among math, science, and English.
Estimate the probability that a randomly selected student from your
school will favor science given that the student is in tenth grade.
Do the same for other subjects and compare the results.
S.CP.5
Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For
example, compare the chance of having lung cancer if you are a
smoker with the chance of being a smoker if you have lung cancer.
Circles
Understand and apply theorems about circles.
G.C.1
Prove that all circles are similar.
G.C.2
Identify and describe relationships among inscribed angles, radii,
and chords. Include the relationship between central, inscribed,
and circumscribed angles; inscribed angles on a diameter are right
angles; the radius of a circle is perpendicular to the tangent where
the radius intersects the circle.
G.C.3
Construct the inscribed and circumscribed circles of a triangle, and
prove properties of angles for a quadrilateral inscribed in a circle.
Find arc lengths and areas of sectors of circles.
G.C.5
Derive using similarity the fact that the length of the arc intercepted
by an angle is proportional to the radius, and define the radian
measure of the angle as the constant of proportionality; derive the
formula for the area of a sector.
Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a
conic section.
G.GPE.1 Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and
radius of a circle given by an equation.
G.GPE.2 Derive the equation of a parabola given a focus and directrix.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given
points in the coordinate plane is a rectangle; prove or disprove that
the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
G.GEP.5 Prove the slope criteria for parallel and perpendicular lines and use
them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given
point).
G.GPE.6 Find the point on a directed line segment between two given points
that partitions the segment in a given ratio.
G.GPE.7 Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.
Geometric Measurement & Dimension
Explain volume formulas and use them to solve problems.
G.GMD.1 Give an informal argument for the formulas for the circumference
of a circle, area of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri’s principle, and informal
limit arguments.
G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres
to solve problems.
Visualize relationships between two-dimensional and three-dimensional
objects.
G.GMD.4 Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
Modeling with Geometry
Apply geometric concepts in modeling situations.
G.MG.1 Use geometric shapes, their measures, and their properties to
describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).
G.MG.3
Apply concepts of density based on area and volume in modeling
situations (e.g., persons per square mile, BTUs per cubic foot).
Apply geometric methods to solve design problems (e.g., designing
an object or structure to satisfy physical constraints or minimize
cost; working with typographic grid systems based on ratios).
Applications of Probability
Use the rules of probability to compute probabilities of compound
events in a uniform probability model.
S.CP.6
Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in terms
of the model.
S.CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.